 Pointwise order on real polynomials

This is a partial order: the real inequalities supply reflexivity and transitivity, while inequalities in both directions give equality of the polynomials as functions. It is not a total order, since the polynomials $x$ and $0$ are incomparable on all of $\mathbb{R}$. Restricting the test domain can change the relation; testing only at finitely many points need not distinguish different polynomials.